## MU02037 Partial Differential Equations I

Mathematical Institute in Opava
Summer 2022
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Jana Kopfová, Ph.D. (lecturer)
doc. RNDr. Jana Kopfová, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Jana Kopfová, Ph.D.
Mathematical Institute in Opava
Prerequisites (in Czech)
TYP_STUDIA ( N )
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives (in Czech)
PDR jsou v jistém smyslu vyvrcholením matematické analýzy, uplatňují se tu výsledky z integrálního a diferenciálního počtu, algebry, geometrie, komplexní analýzy. Přednáška je přehledem klasických výsledků a metod o PDR, budeme se zabývat rovnicemi prvního a druhého řádu.
Syllabus
• 1. Basic notations and definitions. Some known equations. Well posed problems. Short history of PDEs.
2. PDE's of first order, method of characteristics. Linear, quasilinear and nonlinear equations of first order.
3. Cauchy initial problem, Cauchy-Kowalevska theorem.
4. Classification of equations of second order, reduction to the canonical form.
5. Parabolic equations. Derivation of the physical model. Cauchy problem, fundamental solution, properties of solutions of parabolic PDE's, Maximum principle.
6. Fourier's method for parabolic and hyperbolic equations. The Laplace equation on a circle.
7. Hyperbolic equations. Method of characteristics. D'Alembert formula. Hyperbolic equations on a halfline. Three and two-dimensional wave equation. Riemann's method.
8. Elliptic equations. Physical motivation, Harmonic functions. Symmetric solutions, Maximum principle. Potentials: volume potential, simple layer potential, double layer potential. Green's formulas. Dirichlet problem and Neumann problem,
Poisson's formula.
Literature
required literature
• Jan Franců. Parciální diferenciální rovnice. Brno, 1998. info
• M. Renardy, R. C. Rogers. An introduction to partial differential equations. New York, 1993. info
recommended literature
• V. I. Averbuch. Partial differential equations. MÚ SU, Opava. info
• L. C. Evans. Partial diferential equations. 1998. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
Teacher's information
Written exams: 3 written exams during the term, all together for 100 points
For zápočet minumum of 75 points is needed.
Exam: Written exam, oral exam. Student with 90 points or more from the exams during the term has only oral exam.
The course is also listed under the following terms Summer 2021.
• Enrolment Statistics (Summer 2022, recent)